Average Error: 0.1 → 0.1
Time: 13.4s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1} + x}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{\sqrt{1} - x}}{\sqrt[3]{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1} + x}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{\sqrt{1} - x}}{\sqrt[3]{x}}\right)
double f(double x) {
        double r47786 = 1.0;
        double r47787 = x;
        double r47788 = r47786 / r47787;
        double r47789 = r47787 * r47787;
        double r47790 = r47786 - r47789;
        double r47791 = sqrt(r47790);
        double r47792 = r47791 / r47787;
        double r47793 = r47788 + r47792;
        double r47794 = log(r47793);
        return r47794;
}
↓
double f(double x) {
        double r47795 = 1.0;
        double r47796 = x;
        double r47797 = r47795 / r47796;
        double r47798 = sqrt(r47795);
        double r47799 = r47798 + r47796;
        double r47800 = sqrt(r47799);
        double r47801 = cbrt(r47796);
        double r47802 = r47801 * r47801;
        double r47803 = r47800 / r47802;
        double r47804 = r47798 - r47796;
        double r47805 = sqrt(r47804);
        double r47806 = r47805 / r47801;
        double r47807 = r47803 * r47806;
        double r47808 = r47797 + r47807;
        double r47809 = log(r47808);
        return r47809;
}

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-cube-cbrt0.1

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied difference-of-squares0.1

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

  6. Applied sqrt-prod0.1

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

  7. Applied times-frac0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))