Average Error: 0.1 → 0.2
Time: 10.8s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\left(-\log x\right) + \log \left(1 + \sqrt{1 - x \cdot x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\left(-\log x\right) + \log \left(1 + \sqrt{1 - x \cdot x}\right)
double f(double x) {
        double r35547 = 1.0;
        double r35548 = x;
        double r35549 = r35547 / r35548;
        double r35550 = r35548 * r35548;
        double r35551 = r35547 - r35550;
        double r35552 = sqrt(r35551);
        double r35553 = r35552 / r35548;
        double r35554 = r35549 + r35553;
        double r35555 = log(r35554);
        return r35555;
}
↓
double f(double x) {
        double r35556 = x;
        double r35557 = log(r35556);
        double r35558 = -r35557;
        double r35559 = 1.0;
        double r35560 = r35556 * r35556;
        double r35561 = r35559 - r35560;
        double r35562 = sqrt(r35561);
        double r35563 = r35559 + r35562;
        double r35564 = log(r35563);
        double r35565 = r35558 + r35564;
        return r35565;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
  2. Using strategy rm

  3. Applied div-inv0.1

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

  4. Applied div-inv0.1

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

  5. Applied distribute-rgt-out0.1

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

  6. Applied log-prod0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))