Average Error: 32.4 → 0.0
Time: 3.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\sqrt{x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}} \cdot \sqrt{x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\sqrt{x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}} \cdot \sqrt{x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right)
double f(double x) {
        double r82600 = x;
        double r82601 = r82600 * r82600;
        double r82602 = 1.0;
        double r82603 = r82601 - r82602;
        double r82604 = sqrt(r82603);
        double r82605 = r82600 + r82604;
        double r82606 = log(r82605);
        return r82606;
}
↓
double f(double x) {
        double r82607 = x;
        double r82608 = 1.0;
        double r82609 = sqrt(r82608);
        double r82610 = r82607 + r82609;
        double r82611 = sqrt(r82610);
        double r82612 = r82607 - r82609;
        double r82613 = sqrt(r82612);
        double r82614 = r82611 * r82613;
        double r82615 = r82607 + r82614;
        double r82616 = sqrt(r82615);
        double r82617 = r82616 * r82616;
        double r82618 = log(r82617);
        return r82618;
}

Error

Bits error versus x

Try it out

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.4

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

  3. Applied add-sqr-sqrt32.4

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

  4. Applied difference-of-squares32.4

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

  5. Applied sqrt-prod0.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))