Average Error: 32.1 → 0.1
Time: 5.3s
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 r66459 = x;
        double r66460 = r66459 * r66459;
        double r66461 = 1.0;
        double r66462 = r66460 - r66461;
        double r66463 = sqrt(r66462);
        double r66464 = r66459 + r66463;
        double r66465 = log(r66464);
        return r66465;
}
↓
double f(double x) {
        double r66466 = x;
        double r66467 = 1.0;
        double r66468 = sqrt(r66467);
        double r66469 = r66466 + r66468;
        double r66470 = sqrt(r66469);
        double r66471 = r66466 - r66468;
        double r66472 = sqrt(r66471);
        double r66473 = r66470 * r66472;
        double r66474 = r66466 + r66473;
        double r66475 = log(r66474);
        return r66475;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.1

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

  3. Applied add-sqr-sqrt32.1

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

  4. Applied difference-of-squares32.1

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

  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 2020042 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))