Average Error: 32.3 → 0.0
Time: 6.0s
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 r49633 = x;
        double r49634 = r49633 * r49633;
        double r49635 = 1.0;
        double r49636 = r49634 - r49635;
        double r49637 = sqrt(r49636);
        double r49638 = r49633 + r49637;
        double r49639 = log(r49638);
        return r49639;
}
↓
double f(double x) {
        double r49640 = x;
        double r49641 = 1.0;
        double r49642 = sqrt(r49641);
        double r49643 = r49640 + r49642;
        double r49644 = sqrt(r49643);
        double r49645 = r49640 - r49642;
        double r49646 = sqrt(r49645);
        double r49647 = r49644 * r49646;
        double r49648 = r49640 + r49647;
        double r49649 = log(r49648);
        return r49649;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.3

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

  3. Applied add-sqr-sqrt32.3

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

  4. Applied difference-of-squares32.3

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

  5. Applied sqrt-prod0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019212 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))