Average Error: 0.0 → 0.0
Time: 6.1s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \left(\sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}}\right) \cdot \sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \left(\sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}}\right) \cdot \sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}}\right)
double f(double x) {
        double r83924 = 1.0;
        double r83925 = x;
        double r83926 = r83924 / r83925;
        double r83927 = r83925 * r83925;
        double r83928 = r83924 - r83927;
        double r83929 = sqrt(r83928);
        double r83930 = r83929 / r83925;
        double r83931 = r83926 + r83930;
        double r83932 = log(r83931);
        return r83932;
}
↓
double f(double x) {
        double r83933 = 1.0;
        double r83934 = x;
        double r83935 = r83933 / r83934;
        double r83936 = r83934 * r83934;
        double r83937 = r83933 - r83936;
        double r83938 = sqrt(r83937);
        double r83939 = r83938 / r83934;
        double r83940 = cbrt(r83939);
        double r83941 = r83940 * r83940;
        double r83942 = r83941 * r83940;
        double r83943 = r83935 + r83942;
        double r83944 = log(r83943);
        return r83944;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))