Average Error: 0.0 → 0.0
Time: 11.1s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\left(\sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\left(\sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)
double f(double x) {
        double r55961 = 1.0;
        double r55962 = x;
        double r55963 = r55961 / r55962;
        double r55964 = r55962 * r55962;
        double r55965 = r55961 - r55964;
        double r55966 = sqrt(r55965);
        double r55967 = r55966 / r55962;
        double r55968 = r55963 + r55967;
        double r55969 = log(r55968);
        return r55969;
}
↓
double f(double x) {
        double r55970 = 1.0;
        double r55971 = x;
        double r55972 = r55970 / r55971;
        double r55973 = r55971 * r55971;
        double r55974 = r55970 - r55973;
        double r55975 = sqrt(r55974);
        double r55976 = r55975 / r55971;
        double r55977 = r55972 + r55976;
        double r55978 = cbrt(r55977);
        double r55979 = r55978 * r55978;
        double r55980 = r55979 * r55978;
        double r55981 = log(r55980);
        return r55981;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))