Average Error: 31.8 → 0.1
Time: 2.6s
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 r35585 = x;
        double r35586 = r35585 * r35585;
        double r35587 = 1.0;
        double r35588 = r35586 - r35587;
        double r35589 = sqrt(r35588);
        double r35590 = r35585 + r35589;
        double r35591 = log(r35590);
        return r35591;
}
↓
double f(double x) {
        double r35592 = x;
        double r35593 = 1.0;
        double r35594 = sqrt(r35593);
        double r35595 = r35592 + r35594;
        double r35596 = sqrt(r35595);
        double r35597 = r35592 - r35594;
        double r35598 = sqrt(r35597);
        double r35599 = r35596 * r35598;
        double r35600 = r35592 + r35599;
        double r35601 = log(r35600);
        return r35601;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.8

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

  3. Applied add-sqr-sqrt31.8

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

  4. Applied difference-of-squares31.8

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

  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 2020003 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))