Average Error: 0.0 → 0.0
Time: 4.0s
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 r52875 = 1.0;
        double r52876 = x;
        double r52877 = r52875 / r52876;
        double r52878 = r52876 * r52876;
        double r52879 = r52875 - r52878;
        double r52880 = sqrt(r52879);
        double r52881 = r52880 / r52876;
        double r52882 = r52877 + r52881;
        double r52883 = log(r52882);
        return r52883;
}
↓
double f(double x) {
        double r52884 = 1.0;
        double r52885 = x;
        double r52886 = r52884 / r52885;
        double r52887 = r52885 * r52885;
        double r52888 = r52884 - r52887;
        double r52889 = sqrt(r52888);
        double r52890 = cbrt(r52889);
        double r52891 = r52890 * r52890;
        double r52892 = r52890 / r52885;
        double r52893 = r52891 * r52892;
        double r52894 = r52886 + r52893;
        double r52895 = log(r52894);
        return r52895;
}

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