Average Error: 31.7 → 0.1
Time: 10.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left|\sqrt[3]{x + \sqrt{1}}\right| \cdot \sqrt{\sqrt[3]{x + \sqrt{1}}}\right) \cdot \sqrt{x - \sqrt{1}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\left|\sqrt[3]{x + \sqrt{1}}\right| \cdot \sqrt{\sqrt[3]{x + \sqrt{1}}}\right) \cdot \sqrt{x - \sqrt{1}}\right)
double f(double x) {
        double r38545 = x;
        double r38546 = r38545 * r38545;
        double r38547 = 1.0;
        double r38548 = r38546 - r38547;
        double r38549 = sqrt(r38548);
        double r38550 = r38545 + r38549;
        double r38551 = log(r38550);
        return r38551;
}
↓
double f(double x) {
        double r38552 = x;
        double r38553 = 1.0;
        double r38554 = sqrt(r38553);
        double r38555 = r38552 + r38554;
        double r38556 = cbrt(r38555);
        double r38557 = fabs(r38556);
        double r38558 = sqrt(r38556);
        double r38559 = r38557 * r38558;
        double r38560 = r38552 - r38554;
        double r38561 = sqrt(r38560);
        double r38562 = r38559 * r38561;
        double r38563 = r38552 + r38562;
        double r38564 = log(r38563);
        return r38564;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  3. Applied add-sqr-sqrt31.7

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

  4. Applied difference-of-squares31.7

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

  5. Applied sqrt-prod0.1

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

  7. Applied add-cube-cbrt0.1

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

  8. Applied sqrt-prod0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019291 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))