Average Error: 0.0 → 0.3
Time: 3.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\frac{1}{\sqrt[3]{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\frac{1}{\sqrt[3]{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)
double f(double x) {
        double r74788 = 1.0;
        double r74789 = x;
        double r74790 = r74788 / r74789;
        double r74791 = r74789 * r74789;
        double r74792 = r74788 - r74791;
        double r74793 = sqrt(r74792);
        double r74794 = r74793 / r74789;
        double r74795 = r74790 + r74794;
        double r74796 = log(r74795);
        return r74796;
}
↓
double f(double x) {
        double r74797 = 1.0;
        double r74798 = x;
        double r74799 = cbrt(r74798);
        double r74800 = r74799 * r74799;
        double r74801 = r74797 / r74800;
        double r74802 = log(r74801);
        double r74803 = 1.0;
        double r74804 = r74803 / r74799;
        double r74805 = r74798 * r74798;
        double r74806 = r74803 - r74805;
        double r74807 = sqrt(r74806);
        double r74808 = r74807 / r74799;
        double r74809 = r74804 + r74808;
        double r74810 = log(r74809);
        double r74811 = r74802 + r74810;
        return r74811;
}

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

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied times-frac0.0

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

  6. Applied add-cube-cbrt0.1

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

  7. Applied *-un-lft-identity0.1

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

  8. Applied times-frac0.1

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

  9. Applied distribute-lft-out0.1

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

  10. Applied log-prod0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))