Average Error: 32.5 → 0.1
Time: 9.3s
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 r2334428 = x;
        double r2334429 = r2334428 * r2334428;
        double r2334430 = 1.0;
        double r2334431 = r2334429 - r2334430;
        double r2334432 = sqrt(r2334431);
        double r2334433 = r2334428 + r2334432;
        double r2334434 = log(r2334433);
        return r2334434;
}
↓
double f(double x) {
        double r2334435 = x;
        double r2334436 = 1.0;
        double r2334437 = sqrt(r2334436);
        double r2334438 = r2334435 - r2334437;
        double r2334439 = sqrt(r2334438);
        double r2334440 = r2334435 + r2334437;
        double r2334441 = sqrt(r2334440);
        double r2334442 = r2334439 * r2334441;
        double r2334443 = r2334435 + r2334442;
        double r2334444 = log(r2334443);
        return r2334444;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 32.5

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

  3. Applied add-sqr-sqrt32.5

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

  4. Applied difference-of-squares32.5

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

  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 2019171 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))