Average Error: 32.6 → 0.1
Time: 3.4s
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 r83732 = x;
        double r83733 = r83732 * r83732;
        double r83734 = 1.0;
        double r83735 = r83733 - r83734;
        double r83736 = sqrt(r83735);
        double r83737 = r83732 + r83736;
        double r83738 = log(r83737);
        return r83738;
}
↓
double f(double x) {
        double r83739 = x;
        double r83740 = 1.0;
        double r83741 = sqrt(r83740);
        double r83742 = r83739 + r83741;
        double r83743 = sqrt(r83742);
        double r83744 = r83739 - r83741;
        double r83745 = sqrt(r83744);
        double r83746 = r83743 * r83745;
        double r83747 = r83739 + r83746;
        double r83748 = log(r83747);
        return r83748;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.6

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

  3. Applied add-sqr-sqrt32.6

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

  4. Applied difference-of-squares32.6

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

  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 2020034 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))