Average Error: 32.4 → 0.0
Time: 4.0s
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 r78609 = x;
        double r78610 = r78609 * r78609;
        double r78611 = 1.0;
        double r78612 = r78610 - r78611;
        double r78613 = sqrt(r78612);
        double r78614 = r78609 + r78613;
        double r78615 = log(r78614);
        return r78615;
}
↓
double f(double x) {
        double r78616 = x;
        double r78617 = 1.0;
        double r78618 = sqrt(r78617);
        double r78619 = r78616 + r78618;
        double r78620 = sqrt(r78619);
        double r78621 = r78616 - r78618;
        double r78622 = sqrt(r78621);
        double r78623 = r78620 * r78622;
        double r78624 = r78616 + r78623;
        double r78625 = sqrt(r78624);
        double r78626 = r78625 * r78625;
        double r78627 = log(r78626);
        return r78627;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

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