Average Error: 0.0 → 0.0
Time: 4.2s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt[3]{1 - x \cdot x} \cdot \sqrt[3]{1 - x \cdot x}}}{\frac{x}{\sqrt{\sqrt[3]{1 - x \cdot x}}}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt[3]{1 - x \cdot x} \cdot \sqrt[3]{1 - x \cdot x}}}{\frac{x}{\sqrt{\sqrt[3]{1 - x \cdot x}}}}\right)
double f(double x) {
        double r58437 = 1.0;
        double r58438 = x;
        double r58439 = r58437 / r58438;
        double r58440 = r58438 * r58438;
        double r58441 = r58437 - r58440;
        double r58442 = sqrt(r58441);
        double r58443 = r58442 / r58438;
        double r58444 = r58439 + r58443;
        double r58445 = log(r58444);
        return r58445;
}
↓
double f(double x) {
        double r58446 = 1.0;
        double r58447 = x;
        double r58448 = r58446 / r58447;
        double r58449 = r58447 * r58447;
        double r58450 = r58446 - r58449;
        double r58451 = cbrt(r58450);
        double r58452 = r58451 * r58451;
        double r58453 = sqrt(r58452);
        double r58454 = sqrt(r58451);
        double r58455 = r58447 / r58454;
        double r58456 = r58453 / r58455;
        double r58457 = r58448 + r58456;
        double r58458 = log(r58457);
        return r58458;
}

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. Applied associate-/l*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))