Average Error: 0.0 → 0.1
Time: 12.8s
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 r2523738 = 1.0;
        double r2523739 = x;
        double r2523740 = r2523738 / r2523739;
        double r2523741 = r2523739 * r2523739;
        double r2523742 = r2523738 - r2523741;
        double r2523743 = sqrt(r2523742);
        double r2523744 = r2523743 / r2523739;
        double r2523745 = r2523740 + r2523744;
        double r2523746 = log(r2523745);
        return r2523746;
}
↓
double f(double x) {
        double r2523747 = 1.0;
        double r2523748 = x;
        double r2523749 = r2523747 / r2523748;
        double r2523750 = r2523748 * r2523748;
        double r2523751 = r2523747 - r2523750;
        double r2523752 = sqrt(r2523751);
        double r2523753 = r2523752 / r2523748;
        double r2523754 = r2523749 + r2523753;
        double r2523755 = cbrt(r2523754);
        double r2523756 = r2523755 * r2523755;
        double r2523757 = r2523756 * r2523755;
        double r2523758 = log(r2523757);
        return r2523758;
}

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-cbrt0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))