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 r37697 = 1.0;
        double r37698 = x;
        double r37699 = r37697 / r37698;
        double r37700 = r37698 * r37698;
        double r37701 = r37697 - r37700;
        double r37702 = sqrt(r37701);
        double r37703 = r37702 / r37698;
        double r37704 = r37699 + r37703;
        double r37705 = log(r37704);
        return r37705;
}
↓
double f(double x) {
        double r37706 = x;
        double r37707 = log(r37706);
        double r37708 = -r37707;
        double r37709 = 1.0;
        double r37710 = r37706 * r37706;
        double r37711 = r37709 - r37710;
        double r37712 = sqrt(r37711);
        double r37713 = r37709 + r37712;
        double r37714 = log(r37713);
        double r37715 = r37708 + r37714;
        return r37715;
}

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))))