Average Error: 0.0 → 0.0
Time: 3.8s
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 r87180 = 1.0;
        double r87181 = x;
        double r87182 = r87180 / r87181;
        double r87183 = r87181 * r87181;
        double r87184 = r87180 - r87183;
        double r87185 = sqrt(r87184);
        double r87186 = r87185 / r87181;
        double r87187 = r87182 + r87186;
        double r87188 = log(r87187);
        return r87188;
}
↓
double f(double x) {
        double r87189 = 1.0;
        double r87190 = x;
        double r87191 = r87189 / r87190;
        double r87192 = r87190 * r87190;
        double r87193 = r87189 - r87192;
        double r87194 = sqrt(r87193);
        double r87195 = cbrt(r87194);
        double r87196 = r87195 * r87195;
        double r87197 = r87190 / r87195;
        double r87198 = r87196 / r87197;
        double r87199 = r87191 + r87198;
        double r87200 = log(r87199);
        return r87200;
}

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