Average Error: 0.0 → 0.0
Time: 13.4s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\left(\sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\left(\sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)
double f(double x) {
        double r71653 = 1.0;
        double r71654 = x;
        double r71655 = r71653 / r71654;
        double r71656 = r71654 * r71654;
        double r71657 = r71653 - r71656;
        double r71658 = sqrt(r71657);
        double r71659 = r71658 / r71654;
        double r71660 = r71655 + r71659;
        double r71661 = log(r71660);
        return r71661;
}
↓
double f(double x) {
        double r71662 = 1.0;
        double r71663 = x;
        double r71664 = r71662 / r71663;
        double r71665 = r71663 * r71663;
        double r71666 = r71662 - r71665;
        double r71667 = sqrt(r71666);
        double r71668 = r71667 / r71663;
        double r71669 = r71664 + r71668;
        double r71670 = cbrt(r71669);
        double r71671 = r71670 * r71670;
        double r71672 = r71671 * r71670;
        double r71673 = log(r71672);
        return r71673;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied add-cube-cbrt0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))