Average Error: 31.9 → 0.1
Time: 20.8s
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 r70655 = x;
        double r70656 = r70655 * r70655;
        double r70657 = 1.0;
        double r70658 = r70656 - r70657;
        double r70659 = sqrt(r70658);
        double r70660 = r70655 + r70659;
        double r70661 = log(r70660);
        return r70661;
}
↓
double f(double x) {
        double r70662 = x;
        double r70663 = 1.0;
        double r70664 = sqrt(r70663);
        double r70665 = r70662 + r70664;
        double r70666 = sqrt(r70665);
        double r70667 = r70662 - r70664;
        double r70668 = sqrt(r70667);
        double r70669 = r70666 * r70668;
        double r70670 = r70662 + r70669;
        double r70671 = log(r70670);
        return r70671;
}

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 2019322 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))