Average Error: 0.1 → 0.2
Time: 10.1s
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 r50904 = 1.0;
        double r50905 = x;
        double r50906 = r50904 / r50905;
        double r50907 = r50905 * r50905;
        double r50908 = r50904 - r50907;
        double r50909 = sqrt(r50908);
        double r50910 = r50909 / r50905;
        double r50911 = r50906 + r50910;
        double r50912 = log(r50911);
        return r50912;
}
↓
double f(double x) {
        double r50913 = x;
        double r50914 = log(r50913);
        double r50915 = -r50914;
        double r50916 = 1.0;
        double r50917 = r50913 * r50913;
        double r50918 = r50916 - r50917;
        double r50919 = sqrt(r50918);
        double r50920 = r50916 + r50919;
        double r50921 = log(r50920);
        double r50922 = r50915 + r50921;
        return r50922;
}

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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))