Average Error: 0.0 → 0.0
Time: 3.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \sqrt[3]{\sqrt{1 - x \cdot x}}}{\frac{x}{\sqrt[3]{\sqrt{1 - x \cdot x}}}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \sqrt[3]{\sqrt{1 - x \cdot x}}}{\frac{x}{\sqrt[3]{\sqrt{1 - x \cdot x}}}}\right)
double f(double x) {
        double r48074 = 1.0;
        double r48075 = x;
        double r48076 = r48074 / r48075;
        double r48077 = r48075 * r48075;
        double r48078 = r48074 - r48077;
        double r48079 = sqrt(r48078);
        double r48080 = r48079 / r48075;
        double r48081 = r48076 + r48080;
        double r48082 = log(r48081);
        return r48082;
}
↓
double f(double x) {
        double r48083 = 1.0;
        double r48084 = x;
        double r48085 = r48083 / r48084;
        double r48086 = r48084 * r48084;
        double r48087 = r48083 - r48086;
        double r48088 = sqrt(r48087);
        double r48089 = cbrt(r48088);
        double r48090 = r48089 * r48089;
        double r48091 = r48084 / r48089;
        double r48092 = r48090 / r48091;
        double r48093 = r48085 + r48092;
        double r48094 = log(r48093);
        return r48094;
}

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

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

  4. Applied associate-/l*0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))