Average Error: 0.0 → 0.0
Time: 32.1s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{\sqrt{\sqrt[3]{1 - x \cdot x}} \cdot \left|\sqrt[3]{1 - x \cdot x}\right|}{x} + \frac{1}{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{\sqrt{\sqrt[3]{1 - x \cdot x}} \cdot \left|\sqrt[3]{1 - x \cdot x}\right|}{x} + \frac{1}{x}\right)
double f(double x) {
        double r10272427 = 1.0;
        double r10272428 = x;
        double r10272429 = r10272427 / r10272428;
        double r10272430 = r10272428 * r10272428;
        double r10272431 = r10272427 - r10272430;
        double r10272432 = sqrt(r10272431);
        double r10272433 = r10272432 / r10272428;
        double r10272434 = r10272429 + r10272433;
        double r10272435 = log(r10272434);
        return r10272435;
}
↓
double f(double x) {
        double r10272436 = 1.0;
        double r10272437 = x;
        double r10272438 = r10272437 * r10272437;
        double r10272439 = r10272436 - r10272438;
        double r10272440 = cbrt(r10272439);
        double r10272441 = sqrt(r10272440);
        double r10272442 = fabs(r10272440);
        double r10272443 = r10272441 * r10272442;
        double r10272444 = r10272443 / r10272437;
        double r10272445 = r10272436 / r10272437;
        double r10272446 = r10272444 + r10272445;
        double r10272447 = log(r10272446);
        return r10272447;
}

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

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

  4. Applied sqrt-prod0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019119 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))