Average Error: 0.0 → 0.0
Time: 12.2s
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 r657434 = 1.0;
        double r657435 = x;
        double r657436 = r657434 / r657435;
        double r657437 = r657435 * r657435;
        double r657438 = r657434 - r657437;
        double r657439 = sqrt(r657438);
        double r657440 = r657439 / r657435;
        double r657441 = r657436 + r657440;
        double r657442 = log(r657441);
        return r657442;
}
↓
double f(double x) {
        double r657443 = 1.0;
        double r657444 = x;
        double r657445 = r657443 / r657444;
        double r657446 = r657444 * r657444;
        double r657447 = r657443 - r657446;
        double r657448 = sqrt(r657447);
        double r657449 = r657448 / r657444;
        double r657450 = r657445 + r657449;
        double r657451 = cbrt(r657450);
        double r657452 = r657451 * r657451;
        double r657453 = r657452 * r657451;
        double r657454 = log(r657453);
        return r657454;
}

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 2019155 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))