Average Error: 32.3 → 0.0
Time: 3.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{\sqrt[3]{x + \sqrt{1}} \cdot \sqrt[3]{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt[3]{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{\sqrt[3]{x + \sqrt{1}} \cdot \sqrt[3]{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt[3]{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)\right)
double f(double x) {
        double r133431 = x;
        double r133432 = r133431 * r133431;
        double r133433 = 1.0;
        double r133434 = r133432 - r133433;
        double r133435 = sqrt(r133434);
        double r133436 = r133431 + r133435;
        double r133437 = log(r133436);
        return r133437;
}
↓
double f(double x) {
        double r133438 = x;
        double r133439 = 1.0;
        double r133440 = sqrt(r133439);
        double r133441 = r133438 + r133440;
        double r133442 = cbrt(r133441);
        double r133443 = r133442 * r133442;
        double r133444 = sqrt(r133443);
        double r133445 = sqrt(r133442);
        double r133446 = r133438 - r133440;
        double r133447 = sqrt(r133446);
        double r133448 = r133445 * r133447;
        double r133449 = r133444 * r133448;
        double r133450 = r133438 + r133449;
        double r133451 = log(r133450);
        return r133451;
}

Error

Bits error versus x

Try it out

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. Using strategy rm

  7. Applied add-cube-cbrt0.0

    \[\leadsto \log \left(x + \sqrt{\color{blue}{\left(\sqrt[3]{x + \sqrt{1}} \cdot \sqrt[3]{x + \sqrt{1}}\right) \cdot \sqrt[3]{x + \sqrt{1}}}} \cdot \sqrt{x - \sqrt{1}}\right)\]

  8. Applied sqrt-prod0.0

    \[\leadsto \log \left(x + \color{blue}{\left(\sqrt{\sqrt[3]{x + \sqrt{1}} \cdot \sqrt[3]{x + \sqrt{1}}} \cdot \sqrt{\sqrt[3]{x + \sqrt{1}}}\right)} \cdot \sqrt{x - \sqrt{1}}\right)\]

  9. Applied associate-*l*0.0

    \[\leadsto \log \left(x + \color{blue}{\sqrt{\sqrt[3]{x + \sqrt{1}} \cdot \sqrt[3]{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt[3]{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)}\right)\]

  10. Final simplification0.0

    \[\leadsto \log \left(x + \sqrt{\sqrt[3]{x + \sqrt{1}} \cdot \sqrt[3]{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt[3]{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)\right)\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))