Average Error: 0.1 → 0.1
Time: 14.3s
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 r38917 = 1.0;
        double r38918 = x;
        double r38919 = r38917 / r38918;
        double r38920 = r38918 * r38918;
        double r38921 = r38917 - r38920;
        double r38922 = sqrt(r38921);
        double r38923 = r38922 / r38918;
        double r38924 = r38919 + r38923;
        double r38925 = log(r38924);
        return r38925;
}
↓
double f(double x) {
        double r38926 = 1.0;
        double r38927 = x;
        double r38928 = r38926 / r38927;
        double r38929 = sqrt(r38926);
        double r38930 = r38929 + r38927;
        double r38931 = sqrt(r38930);
        double r38932 = cbrt(r38927);
        double r38933 = r38932 * r38932;
        double r38934 = r38931 / r38933;
        double r38935 = r38929 - r38927;
        double r38936 = sqrt(r38935);
        double r38937 = r38936 / r38932;
        double r38938 = r38934 * r38937;
        double r38939 = r38928 + r38938;
        double r38940 = log(r38939);
        return r38940;
}

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