Average Error: 0.0 → 0.0
Time: 3.8s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \left(\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \sqrt[3]{\sqrt{1 - x \cdot x}}\right) \cdot \frac{\sqrt[3]{\sqrt{1 - x \cdot x}}}{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \left(\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \sqrt[3]{\sqrt{1 - x \cdot x}}\right) \cdot \frac{\sqrt[3]{\sqrt{1 - x \cdot x}}}{x}\right)
double f(double x) {
        double r81616 = 1.0;
        double r81617 = x;
        double r81618 = r81616 / r81617;
        double r81619 = r81617 * r81617;
        double r81620 = r81616 - r81619;
        double r81621 = sqrt(r81620);
        double r81622 = r81621 / r81617;
        double r81623 = r81618 + r81622;
        double r81624 = log(r81623);
        return r81624;
}
↓
double f(double x) {
        double r81625 = 1.0;
        double r81626 = x;
        double r81627 = r81625 / r81626;
        double r81628 = r81626 * r81626;
        double r81629 = r81625 - r81628;
        double r81630 = sqrt(r81629);
        double r81631 = cbrt(r81630);
        double r81632 = r81631 * r81631;
        double r81633 = r81631 / r81626;
        double r81634 = r81632 * r81633;
        double r81635 = r81627 + r81634;
        double r81636 = log(r81635);
        return r81636;
}

Error

Bits error versus x

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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 *-un-lft-identity0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied times-frac0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))