Average Error: 0.1 → 0.2
Time: 11.3s
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 r56899 = 1.0;
        double r56900 = x;
        double r56901 = r56899 / r56900;
        double r56902 = r56900 * r56900;
        double r56903 = r56899 - r56902;
        double r56904 = sqrt(r56903);
        double r56905 = r56904 / r56900;
        double r56906 = r56901 + r56905;
        double r56907 = log(r56906);
        return r56907;
}
↓
double f(double x) {
        double r56908 = x;
        double r56909 = log(r56908);
        double r56910 = -r56909;
        double r56911 = 1.0;
        double r56912 = r56908 * r56908;
        double r56913 = r56911 - r56912;
        double r56914 = sqrt(r56913);
        double r56915 = r56911 + r56914;
        double r56916 = log(r56915);
        double r56917 = r56910 + r56916;
        return r56917;
}

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))))