Average Error: 0.1 → 0.1
Time: 9.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \left(\left(\sqrt[3]{\sqrt{\frac{\sqrt{1 - x \cdot x}}{x}}} \cdot \sqrt[3]{\sqrt{\frac{\sqrt{1 - x \cdot x}}{x}}}\right) \cdot \sqrt[3]{\sqrt{\frac{\sqrt{1 - x \cdot x}}{x}}}\right) \cdot \sqrt{\frac{\sqrt{1 - x \cdot x}}{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \left(\left(\sqrt[3]{\sqrt{\frac{\sqrt{1 - x \cdot x}}{x}}} \cdot \sqrt[3]{\sqrt{\frac{\sqrt{1 - x \cdot x}}{x}}}\right) \cdot \sqrt[3]{\sqrt{\frac{\sqrt{1 - x \cdot x}}{x}}}\right) \cdot \sqrt{\frac{\sqrt{1 - x \cdot x}}{x}}\right)
double f(double x) {
        double r82534 = 1.0;
        double r82535 = x;
        double r82536 = r82534 / r82535;
        double r82537 = r82535 * r82535;
        double r82538 = r82534 - r82537;
        double r82539 = sqrt(r82538);
        double r82540 = r82539 / r82535;
        double r82541 = r82536 + r82540;
        double r82542 = log(r82541);
        return r82542;
}
↓
double f(double x) {
        double r82543 = 1.0;
        double r82544 = x;
        double r82545 = r82543 / r82544;
        double r82546 = r82544 * r82544;
        double r82547 = r82543 - r82546;
        double r82548 = sqrt(r82547);
        double r82549 = r82548 / r82544;
        double r82550 = sqrt(r82549);
        double r82551 = cbrt(r82550);
        double r82552 = r82551 * r82551;
        double r82553 = r82552 * r82551;
        double r82554 = r82553 * r82550;
        double r82555 = r82545 + r82554;
        double r82556 = log(r82555);
        return r82556;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

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

  3. Applied add-sqr-sqrt0.1

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

  5. Applied add-cube-cbrt0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))